Chaos in the fractional order Chen system and its control. (English) Zbl 1069.37025

Summary: We study the chaotic behavior in the fractional order Chen system. We show that chaos exists in the fractional order Chen system with order less than 3. The lowest order we found to have chaos in this system is 2,1. Linear feedback control of chaos in this system is also studied.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
26A33 Fractional derivatives and integrals
93B52 Feedback control
Full Text: DOI


[1] Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[2] (Hilfer, R., Applications of fractional calculus in physics (2001), World Scientific: World Scientific New Jersey) · Zbl 0998.26002
[3] Bagley, R. L.; Calico, R. A., Fractional order state equations for the control of viscoelastically damped structures, J. Guid., Contr. Dyn., 14, 304-311 (1991)
[4] Koeller, R. C., Application of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51, 299 (1984) · Zbl 0544.73052
[5] Koeller, R. C., Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics, Acta Mech., 58, 251-264 (1986) · Zbl 0578.73040
[6] Sun, H. H.; Abdelwahad, A. A.; Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE Trans. Auto. Contr., 29, 441-444 (1984) · Zbl 0532.93025
[7] Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of noninteger order transfer functions for analysis of electrode process, J. Electroanal. Chem., 33, 253-265 (1971)
[8] Heaviside, O., Electromagnetic theory (1971), Chelsea: Chelsea New York
[9] Oustaloup, A.; Sabatier, J.; Lanusse, P., From fractal robustness to CRONE control, Fract. Calculus Appl. Anal., 2, 1-30 (1999) · Zbl 1111.93310
[10] Oustaloup, A.; Levron, F.; Nanot, F.; Mathieu, B., Frequency band complex non integer differentiator: characterization and synthesis, IEEE Trans. CAS-I, 47, 25-40 (2000)
[11] Chen, Y. Q.; Moore, K., Discretization schemes for fractional-order differentiators and integrators, IEEE Trans. CAS-I, 49, 363-367 (2002) · Zbl 1368.65035
[12] Hartley, T. T.; Lorenzo, C. F., Dynamics and control of initialized fractional-order systems, Nonlinear Dyn., 29, 201-233 (2002) · Zbl 1021.93019
[13] Hwang, C.; Leu, J.-F.; Tsay, S.-Y., A note on time-domain simulation of feedback fractional-order systems, IEEE Trans. Auto. Contr., 47, 625-631 (2002) · Zbl 1364.93772
[14] Podlubny, I.; Petras, I.; Vinagre, B. M.; O’Leary, P.; Dorcak, L., Analogue realizations of fractional-order controllers, Nonlinear Dyn., 29, 281-296 (2002) · Zbl 1041.93022
[15] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K., Chaos in a fractional order Chua’s system, IEEE Trans. CAS-I, 42, 485-490 (1995)
[16] Arena P, Caponetto R, Fortuna L, Porto D. Chaos in a fractional order Duffing system. In: Proc. ECCTD, Budapest 1997. p. 1259-62; Arena P, Caponetto R, Fortuna L, Porto D. Chaos in a fractional order Duffing system. In: Proc. ECCTD, Budapest 1997. p. 1259-62
[17] Ahmad, W.; El-Khazali, R.; El-Wakil, A., Fractional-order Wien-bridge oscillator, Electr. Lett., 37, 1110-1112 (2001)
[18] Ahmad, W. M.; Sprott, J. C., Chaos in fractional-order autonomous nonlinear systems, Chaos, Solitons & Fractals, 16, 339-351 (2003) · Zbl 1033.37019
[19] Ahmad, W. M.; Harb, W. M., On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos, Solitons & Fractals, 18, 693-701 (2003) · Zbl 1073.93027
[20] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 91, 034101 (2003)
[21] Arena, P.; Caponetto, R.; Fortuna, L.; Porto, D., Bifurcation and chaos in noninteger order cellular neural networks, Int. J. Bifur. Chaos, 7, 1527-1539 (1998) · Zbl 0936.92006
[22] Arena, P.; Fortuna, L.; Porto, D., Chaotic behavior in noninteger-order cellular neural networks, Phys. Rev. E, 61, 776-781 (2000)
[23] Li C, Chen G. Chaos and hyperchaos in fractional order Rössler equations. Preprint, 2003; Li C, Chen G. Chaos and hyperchaos in fractional order Rössler equations. Preprint, 2003
[24] Li, C.; Liao, X.; Yu, J., Synchronization of fractional order chaotic systems, Phys. Rev. E, 68, 067203 (2003)
[25] Chen, G.; Ueta, T., Yet another chaotic attractor, Int. J. Bifur. Chaos, 9, 1465-1466 (1999) · Zbl 0962.37013
[26] Charef, A.; Sun, H. H.; Tsao, Y. Y.; Onaral, B., Fractal system as represented by singularity function, IEEE Trans. Auto. Contr., 37, 1465-1470 (1992) · Zbl 0825.58027
[27] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990) · Zbl 0964.37501
[28] (Chen, G.; Yu, X., Chaos control: theory and applications (2003), Springer-Verlag: Springer-Verlag New York) · Zbl 1029.00015
[29] Chen, G.; Dong, X., From chaos to order: methodologies, perspectives and applications (1998), World Scientific: World Scientific Singapore · Zbl 0908.93005
[30] Li C. Private communication with Grigorenko I. 2003; Li C. Private communication with Grigorenko I. 2003
[31] Zaslavsky, G. M., Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371, 461-580 (2002) · Zbl 0999.82053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.